A Further Note on the Stable Matching Problem, Gabrielle Demange, David Gale, and Marilda Sotomayor, Discrete Applied Mathematics, 1987 (limits to manipulation)

A Non-constructive Elementary Proof of the Existence of Stable Marriages, Marilda Sotomayor, Games and Economic Behavior, 1996 (structure, still to come?)

I pointed out that some theory is fundamental because it plays a big role in producing new theory, and understanding mathematical structure, and some theory is fundamental because it helps us understand the world, make sense of empirical observations, and explains how and why successful market designs work.

In the latter category, I've come to appreciate the theorem about the limits to which stable matching mechanisms can be manipulated by misrepresenting preferences in simple matching markets:

Limits on successful manipulation (Demange,
Gale, and Sotomayor). Let P be the true preferences (not necessarily strict) of
the agents, and let P
differ from P in that some coalition C
of men and women misstate their preferences. Then there is no matching m, stable for P,
which is preferred to every stable matching under the true
preferences P by all members of C

Taken together with the variety of empirical and theoretical observations that say that the set of stable matchings is generally quite small, this result states that in general very few agents will be in a position to profitably manipulate their preferences. So it can be viewed as providing an explanation of why stable matching mechanisms have succeeded so well empirically, despite the theoretical result in Roth (1982) stating that no stable matching mechanism can always make it a dominant strategy for agents to reveal their true preferences.